In reply to marra:
We stayed at Bluebell Farm in Belford last week:
http://www.bluebellfarmbelford.com/
Good location - effectively in the village (with Co-Op, pubs, chippy), near Bamburgh, Berwick etc. Decent site, well maintained, it's small and mostly full of static caravans but they didn't seem to have the uptight, petty mentality that often goes with that. Dogs seemed to be welcome although the place has a massive population of semi-tame rabbits, ducks, swans etc so possibly not so great if your dog tends to chase stuff.
The tourist information in Alnwick had a massive list of campsites for the whole Northumberland coast, but they don't seem to put it on their website.
Oh, FWIW (and sorry if you already know this) we thought that actually going in to Bamburgh Castle was a bit of a waste of time and money - it looks amazing from the outside, but the interior is mostly just endless collections of boring Victoriana with fairly little to interest any normal person. (Lindisfarne Castle, by contrast, is an interesting and informative collection of high class Edwardiana and is great if you like that sort of thing aren't expecting dungeons and swords.)